Binary Multiplication

• Initial values:
  • Multiplicand ($n$): given with LHS 0’s
  • Multiplier ($n$): given with RHS 0’s
  • Product ($2n$): 0
• Multiplier LSB dictates whether the multiplicand is added to product
• End after $n$ steps

Binary Division

• Initial values:
  • Quotient ($n$): 0
  • Divisor ($n$): given with RHS 0’s
  • Remainder ($2n$): dividend
• $R-D$ determines the LSB of $Q$ with each step
• End after $n+1$ steps

IEEE 754 Standard

• Sign bit
• Significand:
  • (1 + Mantissa)
  • Mantissa is unsigned 24/53-bit fractional value
• Exponent:
  • (Power - Bias)
  • Power is unsigned 8/11-bit
  • Bias is fixed at 127/1023
    • Introduced so that floats could be sorted without any ambiguity with respect to the 2’s complement representation of the Power

• Steps for problem solving:
  • Convert the base-10 representation to binary fractional representation
  • Convert to normalized form
  • Figure out mantissa and power

Special Cases

• Special cases for which the exponents zero and all ones are reserved
• If $M$ and $P$ zero, the number is zero
• If $M=0$ and $P=11111\dots$, the number is $\pm \infty$
• If $M \ne 0$ and $P=1111\dots$, the number is $\text{NaN}$

References

• Course slides, chapter 3